The use of the local likelihood method (Tibshirani and Hastie, 1987; Loader, 1996) in the presence of interval-censored or aggregated data leads to a natural consideration of an EM-type strategy, or rather a local EM algorithm. In the thesis, we consider local EM to analyze the point process data that are either interval-censored or aggregated into regional counts. We specifically formulate local EM algorithms for density, intensity and risk estimation and implement the algorithms using a piecewise constant function. We demonstrate that the use of the piecewise constant function at the E-step explicitly results in an iteration that involves an expectation, maximization and smoothing step, or an EMS algorithm considered in Silverman, Jones, Wilson and Nychka (1990). Consequently, we reveal a previously unknown connection between local EM and the EMS algorithm. From a theoretical perspective, local EM and the EMS algorithm complement each other. Although the statistical methodology literature often characterizes EMS methods as ad hoc, local likelihood suggests otherwise as the EMS algorithm arises naturally from a local likelihood consideration in the context of point processes. Moreover, the EMS algorithm not only serves as a convenient implementation of the local EM algorithm but also provides a set of theoretical tools to better understand the role of local EM. In particular, we present results that reinforce the suggestion that the pair of local EM and penalized likelihood are analogous to that of EM and likelihood. Applications include the analysis of bivariate interval-censored data as well as disease mapping for a rare disease, lupus, in the Greater Toronto Area.

The use of the local likelihood method (Tibshirani and Hastie, 1987; Loader, 1996) in the presence of interval-censored or aggregated data leads to a natural consideration of an EM-type strategy, or rather a local EM algorithm. In the thesis, we consider local EM to analyze the point process data that are either interval-censored or aggregated into regional counts. We specifically formulate local EM algorithms for density, intensity and risk estimation and implement the algorithms using a piecewise constant function. We demonstrate that the use of the piecewise constant function at the E-step explicitly results in an iteration that involves an expectation, maximization and smoothing step, or an EMS algorithm considered in Silverman, Jones, Wilson and Nychka (1990). Consequently, we reveal a previously unknown connection between local EM and the EMS algorithm. From a theoretical perspective, local EM and the EMS algorithm complement each other. Although the statistical methodology literature often characterizes EMS methods as ad hoc, local likelihood suggests otherwise as the EMS algorithm arises naturally from a local likelihood consideration in the context of point processes. Moreover, the EMS algorithm not only serves as a convenient implementation of the local EM algorithm but also provides a set of theoretical tools to better understand the role of local EM. In particular, we present results that reinforce the suggestion that the pair of local EM and penalized likelihood are analogous to that of EM and likelihood. Applications include the analysis of bivariate interval-censored data as well as disease mapping for a rare disease, lupus, in the Greater Toronto Area.